Ratio, Proportion and Percentages Formulas and Tricks

Ratio & Proportion

  • The equality of two ratios is called a proportion. If a:b=c:d , we write a:b::c:d and we say that a, b, c, d are in proportion. In a proportion, the first and fourth terms are known as extremes, while the second and third are known as means.

  • Productofextremes=Productofmeans

  • Mean proportion between a and b is

  • The compounded ratio of the ratios (a:b),(c:d),(e:f) is (ace:bdf)

  • a2:b2 is a duplicate ratio of a:b

  • a:b is a sub-duplicate ration of a:b

  • a3:b3 is a triplicate ratio of a:b

  • a1/3:b1/3 is a sub-triplicate ratio of a:b

  • If a/b=c/d , then ,(a+b)/b=(c+d)/d , which is called the Componendo.

  • If a/b=c/d,then,(ab)/b=(cd)/d , which is called the dividendo

  • If a/b=c/d,then,(a+b)/(ab)=(c+d)/(cd) , which is called the componendo & dividendo.

  • Variation: We say that x is directly proportional to y if x=ky for some constant k and we write, xαy.

  • Also, we say that x is inversely proportional to y ifx=k/y for some constant k and we write xα1/y.

Ratios

  • Ifa:b=c:d,thena:b=c:d=(a+c):(b+d)

  • If a<b, then for a positive quantity x,

  • a+xb+x>abandaxbx<ab If a>b, then for a positive quantity x,

  • a+xb+x<abandaxbx>ab

Proportions

Ifa:bc:dorab=cd,then

  • ac=bdAlternendoLaw

  • ba=dcInvertendoLaw

  • a+bb=c+ddComponendoLaw

  • abb=cddDividendoLaw

  • a+bab=c+dcdComponendoandDividendoLaw

  • Ifab=cd=ef=k,thena+c+e+b+d+f+=k

  • Ifab=cd=ef=k,thenp,q,rarerealnumbers,thenpan+qcn+ren+pbn+qdn+rfn+=kn

Percentage

  • To express x% as a fraction, we have x%=x/100

  • To express a/b as a percent, we have a/b=(a/b×100)%

  • If ‘A’ is R% more than ‘B’, then ‘B’ is less than ‘A’ by

    OR

    If the price of a commodity increases by R%, then the reduction in consumption, not to increase the expenditure is {100R/[100+R]}%

  • If ‘A’ is R% less than ‘B’, then ‘B’ is more than ‘A’ by

    OR

    If the price of a commodity decreases by R%, then the increase in consumption, not to increase the expenditure is {100R/[100R]}%

  • If the population of a town is ‘P’ in a year, then its population after ‘N’ years is P(1+R/100)N

  • If the population of a town is ‘P’ in a year, then its population ‘N’ years ago is P/[(1+R/100)N]

Percentage Change

  • PercentageChange=FinalValueInitialValueInitialValue×100

  • For two successive changes of a% and b%, TotalPercentageChange=(a+b+ab100)%